Fake diagram lemmas that look like they should work but don't.

Question

I got a book on homological algebra in a textbook giveaway and I'm just starting to learn more about exact sequences in preparation for reading the book more seriously. I have seen things like the following before but don't understand them:

• The snake lemma
• The five lemma
• The nine lemma

These lemmas, in my limited understanding, show you how to propagate exactness conditions through a commutative diagram or infer additional properties of some morphism somewhere.

These lemmas are hard to appreciate, though, without some examples in mind of what can go wrong.

So, I'm wondering if there are any instructive examples of fake diagram lemmas that are wrong for a subtle reason.

Answer 1

If "fake" diagram lemmas fail, it likely won't be for a subtle reason, but rather just because there is a counterexample.

Anyway, here is one example: given a commutative diagram of exact sequences $$ \require{AMScd} \begin{CD} 0 @>>> A @>>> B @>>> C @>>> 0 \\ @. @V{0}VV @V{f}VV @V{0}VV\\ 0 @>>> A' @>>> B' @>>> C' @>>> 0 \\ \end{CD} $$ it need not be the case that $f=0$.